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On the $K(\unicode[STIX]{x1D70B},1)$-problem for restrictions of complex reflection arrangements

Part of: Special aspects of infinite or finite groups Projective and enumerative geometry Discrete geometry Differential algebra

Published online by Cambridge University Press:  20 January 2020

Nils Amend ,
Pierre Deligne  and
Gerhard Röhrle
Nils Amend
Affiliation:
Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Fakultät für Mathematik und Physik, Gottfried Wilhelm Leibniz Universität Hannover, Welfengarten 1, D-30167 Hannover, Germany email amend@math.uni-hannover.de
Pierre Deligne
Affiliation:
Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA email deligne@math.ias.edu
Gerhard Röhrle
Affiliation:
Fakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstraße 150, D-44780 Bochum, Germany email gerhard.roehrle@rub.de
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Abstract

Let $W\subset \operatorname{GL}(V)$ be a complex reflection group and $\mathscr{A}(W)$ the set of the mirrors of the complex reflections in $W$. It is known that the complement $X(\mathscr{A}(W))$ of the reflection arrangement $\mathscr{A}(W)$ is a $K(\unicode[STIX]{x1D70B},1)$ space. For $Y$ an intersection of hyperplanes in $\mathscr{A}(W)$, let $X(\mathscr{A}(W)^{Y})$ be the complement in $Y$ of the hyperplanes in $\mathscr{A}(W)$ not containing $Y$. We hope that $X(\mathscr{A}(W)^{Y})$ is always a $K(\unicode[STIX]{x1D70B},1)$. We prove it in case of the monomial groups $W=G(r,p,\ell )$. Using known results, we then show that there remain only three irreducible complex reflection groups, leading to just eight such induced arrangements for which this $K(\unicode[STIX]{x1D70B},1)$ property remains to be proved.

Keywords

complex reflection groupsreflection arrangementsrestricted arrangementsK (𝜋, 1)-arrangementsEilenberg–MacLane space

MSC classification

Primary: 20F55: Reflection and Coxeter groups 52C35: Arrangements of points, flats, hyperplanes 14N20: Configurations and arrangements of linear subspaces
Secondary: 13N15: Derivations
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 3 , March 2020 , pp. 526 - 532
Copyright
© The Authors 2020

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